No, I don't see. Precise communication is of the utmost importance in life and death situations. Consider pilots. The communication is extremely structured so as to precisely convey relevant information. Pilots are trained specifically and at length on how to communicate over radios in normal and emergency situations. This is done to make communication more efficient and effective. And these choices are data driven.
You're arguing for essentially the opposite of that.
> They're relevant details because they directly affect the health and safety of the ambulance staff
A person being homeless is not relevant to the health and safety of the medical staff. A person sitting on wet cardboard is not relevant to the health and safety of the medical staff. The area being hazardous is relevant, and was conveyed. You may argue that "the area has feces and needles" is necessary to communicate. And that's fine.
> This is why speech must be free, because I don't want a paramedic who's treating me to be debating exactly how to define me to dispatch
No, if this is your goal what you want is paramedics who are well trained in how to communicate effectively over radios, much like pilots are. This has nothing to do with free speech. Restrictions and regimentation on speech in this context would actually make the communication more effective. For example, specific guidance on what things should be communicated about the hazardous area: that feces and needles are relevant but that cardboard boxes and homelessness aren't. This way, there are fewer wasted words.